On the Distributions of the State Sizes of Discrete Time Homogeneous Markov Systems

  • G. Vasiliadis
  • G. Tsaklidis


The evolution of a closed discrete-time homogeneous Markov system (HMS) is determined by the evolution of its state sizes in time. In order to examine the variability of the state sizes, their moments are evaluated for any time point, and recursive formulae for their computation are derived. As a consequence the asymptotic values of the moments for a convergent HMS can be evaluated. The respective recursive formula for a HMS with periodic transition matrix is given. The p.d.f.’s of the state sizes follow directly by means of the moments. The theoretical results are illustrated by a numerical example.


Stochastic population systems Discrete-time homogeneous Markov models Markov systems 

AMS 2000 Subject Classification

Primary 90B70 62E99; Secondary 91D35 


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  1. D. J. Bartholomew, Stochastic Models for Social Processes. 3rd edn, Wiley: New York, 1982.MATHGoogle Scholar
  2. J. Gani, “Formulae for projecting enrolments and degrees awarded in universities,” Journal of the Royal Statistical Society. Series A. General vol. 126 pp. 400–409, 1963.CrossRefGoogle Scholar
  3. D. L. Isaacson and R. W. Madsen, Markov Chains. Theory and Applications, Wiley: New York, 1976.MATHGoogle Scholar
  4. I. Kipouridis and G. Tsaklidis, “The size order of the state vector of discrete-time homogeneous Markov systems,” Journal of Applied Probability vol. 38 pp. 357–368, 2001.MATHCrossRefMathSciNetGoogle Scholar
  5. S. I. McClean, B. McAlea, and P. Millard, “Using a Markov reward model to estimate spend-down costs for a geriatric department,” Journal of the Operational Research Society vol. 10 pp. 1021-1025, 1998.CrossRefGoogle Scholar
  6. P. D. Patoucheas and G. Stamou, “Non-homogeneous Markovian models in ecological modelling: a study of the zoobenthos dynamics in Thermaikos Gulf, Greece,” Ecological Modelling vol. 66 pp. 197–215, 1993.CrossRefGoogle Scholar
  7. G. J. Taylor, S. I. McClean, and P. Millard, “Stochastic model of geriatric patient bed occupancy behaviour,” Journal of the Royal Statistical Society vol. 163(1) pp. 39–48, 2000.CrossRefGoogle Scholar
  8. G. Tsaklidis, “The evolution of the attainable structures of a homogeneous Markov system with fixed size,” Journal of Applied Probability vol. 31 pp. 348–361, 1994.MATHCrossRefMathSciNetGoogle Scholar
  9. G. Tsaklidis and K. P. Soldatos, “Modelling of continuous time homogeneous Markov system with fixed size as elastic solid,” Applied Mathematical Modelling vol. 27 pp. 877–887, 2003.MATHCrossRefGoogle Scholar
  10. P.-C. G. Vassiliou, “Asymptotic behaviour of Markov systems,” Journal of Applied Probability vol. 19 pp. 851–857, 1982.MATHCrossRefMathSciNetGoogle Scholar
  11. P.-C. G. Vassiliou, “The evolution of the theory of non-homogeneous Markov systems,” Applied Stochastic Models and Data Analysis vol. 13 no. 3–4 pp. 159–176, 1997.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsAristotle University of ThessalonikiThessalonikiGreece

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